Cours de Jean-Pierre Serre, no. 1 (1981) , 204 p.

### Sommaire

 History p. 1 Siegel formula p. 3 Tamagawa p. 8 I - Integration on $p$-adic manifolds page p. 9 Notation p. 9 Measure attached to a form $\omega$ p. 10 Rational summation p. 15 Igusa p. 16 Vector bundles p. 17 Smooth schemes p. 18 Number of points of classical and exceptional groups p. 20 Decomposition of a measure p. 24 Connection with densities p. 28 Digression : Liftable solutions p. 31 Smooth case p. 32 Real case p. 37 Use of resolution of singularities p. 40 Oesterlé : hypersurfaces p. 41 Łojasiewicz inequality p. 48 Oesterlé : general case p. 49 II - Adeles p. 55 History p. 55 Definition p. 55 $\mathbb{A}_K/K$ compact p. 58 Haar measure on $\mathbb{A}_K$ p. 61 Characters and duality p. 64 Haar measure for dual groups p. 66 Compatible measures with respect to duality p. 67 Adelic integration and heuristic formulas (Goldbach,...) p. 69 Adelic points of algebraic varieties p. 75 Properties of the functor $V\mapsto V(\mathbb{A}_K)$ p. 78 Restriction of scalars p. 84 Algebraic groups and adelic points p. 88 Abelian varieties (S. Bloch) p. 88 Weak approximation p. 90 Strong approximation p. 91 Adeles, classes and genera p. 99 Tensors page p. 102 Vector bundles p. 104 Adelic measures p. 106 Convergent case p. 108 Algebraic groups p. 111 Tori p. 114 Convergence p. 117 Tamagawa number (semi-simple case) p. 122 Theorem of Ono – Weil conjecture p. 123 Tamagawa number (reductive case) p. 126 $\tau(\mathbb{G}_m)=1$ p. 131 Tori (Ono) p. 133 $\tau(PGL_n)=n \tau(SL_n)$ p. 135 Tamagawa $\leftrightarrow$ Siegel (mass formula) p. 136 Tamagawa $\leftrightarrow$ Siegel (the two-groups game) p. 145 Correction to Ono's theory p. 146 Positive definite quadratic forms over $\mathbb{Q}$ p. 147 Proof that $\tau(SO_n)=2$ for $n=2,3,4$ p. 152 Proof of Siegel's formula p. 156 Proof ($m\geq 5$) p. 163 Proof ($m=4$) p. 165 Proof ($m=2$) p. 166 Proof ($m=3$) p. 169 Remarks on Siegel's proof p. 172 Application to modular forms p. 176 III - $SL_n$ p. 180 The Minkowski-Hlawka theorem p. 180 Proof p. 184

@book{CJPS_1981__1_,
author = {Serre, Jean-Pierre},
title = {Adeles and {Tamagawa} numbers},
series = {Cours de Jean-Pierre Serre},
publisher = {Harvard},
number = {1},
year = {1981},
language = {en},
url = {http://www.numdam.org/item/CJPS_1981__1_/}
}
TY  - BOOK
AU  - Serre, Jean-Pierre
TI  - Adeles and Tamagawa numbers
T3  - Cours de Jean-Pierre Serre
PY  - 1981
DA  - 1981///
IS  - 1
PB  - Harvard
UR  - http://www.numdam.org/item/CJPS_1981__1_/
LA  - en
ID  - CJPS_1981__1_
ER  - 
Serre, Jean-Pierre. Adeles and Tamagawa numbers. Cours de Jean-Pierre Serre, no. 1 (1981), 204 p. http://numdam.org/item/CJPS_1981__1_/